Caustic (mathematics)

In differential geometry a caustic is the envelope of rays either reflected or refracted by a manifold. Obviously it is related to the optical concept of caustics.

The ray's source may be a point (called the radiant) or infinity, in which case a direction vector must be specified.

Catacaustic

A catacaustic is the reflective case.

With a radiant, it is the evolute of the orthotomic of the radiant.

The planar, parallel-source-rays case: suppose the direction vector is <math>(a,b)<math> and the mirror curve is parametrised as <math>(u(t),v(t))<math>. The normal vector at a point is <math>(-v'(t),u'(t))<math>; the reflection of the direction vector is

<math>2\mbox{proj}_nd-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{

(av'^2-2bu'v'-au'^2,bu'^2-2au'v'-bv'^2) }{v'^2+u'^2}<math> so the reflected ray satisfies

<math>(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2)<math>

Using the simplest envelope form

<math>F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2)<math> <math>=x(bu'^2-2au'v'-bv'^2)

-y(av'^2-2bu'v'-au'^2) +b(uv'^2-uu'^2-2vu'v') +a(-vu'^2+vv'^2+2uu'v')<math>

<math>F_t(x,y,t)=2x(bu'u''-a(u'v''+u''v')-bv'v'')

-2y(av'v''-b(u''v'+u'v'')-au'u'') +b( u'v'^2 +2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'') +a(-v'u'^2 -2vu'u'' +v'^3 +2vv'v'' +2v'u'^2 +2v''uu' +2v'uu'')<math> which looks horrid, but <math>F=F_t=0<math> gives a linear system in <math>(x,y)<math> and so it is elementary to obtain a parametrisation of the catacaustic. Cramer's rule would serve.

Example

Let the direction vector be (0,1) and the mirror be <math>(t,t^2)<math> Then

<math>u'=1<math>   <math>u''=0<math>   <math>v'=2t<math>   <math>v''=2<math>   <math>a=0<math>   <math>b=1<math>

<math>F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t<math>

<math>F_t(x,y,t)=-8tx+4y-1<math>

and <math>F=F_t=0<math> has solution <math>(0,1/4)<math>; i.e., light entering a parabolic mirror parallel to its axis is reflected through the focus.

Diacaustic

A diacaustic is the refractive case. It is complicated by the need for another datum (refractive index) and the fact refraction is not linear -- Snell's law is "ugly" in pure vector notation.